NASA-CR-193727

SPATIALLY-VARYING IIR FILTER BANKS FOR IMAGE CODING

Wilson C. Chung and Mark J. T. Smith

School of Electrical EngineeringGeorgia Institute of Technology

Atlanta, GA 30332

ABSTRACTThis paper reports on the application of spatially variantIIR filter banks to subband image coding. The new filterbank is based on computationally efficient recursive poly-phase decompositions that dynamically change in responseto the input signal. In the absence of quantization, recon-struction can be made exact. However, by proper choice ofan adaptation scheme, we show that subband image codingbased on time varying filter banks can yield improvementover the use of conventional filter banks.

1. INTRODUCTION

Recursive analysis and synthesis filter banks for subbandcoding have been considered previously and were shown tobe particularly attractive because they preserve exact re-construction and can achieve tremendous computational ef-ficiency relative to comparable FIR systems [6], [8]. Thesefilter banks are typically two-band systems that serve asbuilding blocks for uniform and non-uniform band tree-structured systems.

Much attention has been given in minimizing the per-ceptually significant coding distortions in image coding sys-tems. At low rates, aliasing and ringing distortions tendto be pronounced and have sperned some efforts to reducethese effects; but the success has been limited [9], [6]. Thedifficulty is that the step response ripples of the niters,which are the source of the ringing distortion, are neededto achieve good filter magnitude response characteristics.Good spectral magnitude characteristics are important forreducing the effects of aliasing in the presence of coding.Aliasing distortion appears as blurring in the image and issubjectively noticeable. If filters with good magnitude char-acteristics are used to reduce the aliasing, then ringing dis-tortion results which is also very noticeable and objection-able. Since both good step response and good magnituderesponse characteristics can not be achieved simultaneouslyfor conventional filter banks, this poses a dilemma.

A solution to this problem using IIR time-varying fil-ter banks was proposed at the IEEE DSP Workshop [2]and a theoretical analysis of the filter bank was presentedin [4]. In this paper, we expand upon this idea giving spe-cial attention to its application to image coding. In thenext section, we briefly summarize the results derived in [4]

ro[n]

This work is supported in part by the National Science Foun-dation under contract MIP-9116113 and the National Aeronau-tics and Space Administration.

*(»)-_^

z-1 , |

12

12r,M

Po(z)

—•]*(*)

•"I'M

,,Mx/ , N. . . r_-i

* * »HnJ

(a) Analysis

»[•»;L

X1 1 \

Qo(«)

Q:W

T 2

T 2 - ^ J-*(«)

(b) Synthesis

Figure 1: Two-Band Polyphase Implementation of theAnalysis/Synthesis System (a) Analysis, (b) Synthesis

which are the basis for this work. In the follow section weconsider a control mechanism for performing the adapta-tion and compare the performance of the overall system tothat of a conventional system.

2. SPATIALLY-VARIANT FILTER BANKS

The two-band polyphase system, as shown in Figure 1, iscommonly used in conventional subband analysis/synthesissystems. The lowpass and highpass analysis filters, Ho(z)tand H\(z), have the form:

H0(z) = (1)(2)

where Po(z) and PI(Z) are the analysis polyphase filters andQo(z) and Qi(z) are the synthesis polyphase filters. Thissystem has the property that its structure guarantees exact

(NASA-CR-193727) SPATIALLY-VARYINGIIR FILTER BANKS FOR IMAGE CODING(Georgia Inst. of Tech.) 4 p

N94-12897

Unclas

https://ntrs.nasa.gov/search.jsp?R=19940008424 2020-04-20T20:03:53+00:00Z

reconstruction when for 0 < n < oo and

0.0) = and

The polyphase filters, Po(z) and PI (2), are assumed to becausal stable recursive niters with zeros strictly outside theunit circle and have allpass or near allpass characteristics.The synthesis polyphase filters, Qo(z) and Qi(z), by defi-nition will have their poles outside the unit circle and theirzeros inside the circle. Consequently they are anti-causaland stable.

The new spatially-varying system is identical exceptthat the filters change in response to the input. In particu-lar, the coefficients of the polyphase filters Po(«) and Pi(z)are changed selectively at some point in the filtering processto another set of polyphase filters. Thus to preserve exactreconstruction it is sufficient to guarantee that rj[n] = f;[n](where i =s 0,1) at all times before, during, and after theswitching of coefficients. The derivation presented in [4]considered starting with a set of polyphase filters (pn'fn],pf [n]) and switching them to another set of filters (PO ["].Pi [n]). For convenience, assume that we switch the filtersat n = 0 and that the polyphase filters have the form:

-f a^z'1 + • L -MO

*-» -I-

a* -I- of 2"1 + •

and

The labels L and R signify that the signal or filter coefficientis associated with the left and right half of the time indexrespectively.

The direct form difference equations that implement thepolyphase filters are:

Mo No

in the range —oo < n < —1 and

M, JVj

(3)

(4)msO

when 0 < n < oo. Note that ta[n] and »«[n] are the left-sided and right-sided components of a polyphase filter out-put »[n], i.e. v[n] = ta[n] + vji[n]. Equation (4) requires theinitial conditions VR[— Ni], VR[~NI -1], ..., VR[—!] in or-der to evaluate VR[O]. These initial conditions are obtainedfrom the past samples of the output, i.e. VR[n] = f/,[n] inthe range -Ni <n< —1.

The reconstruction equations (which are derived in [4])are performed by anti-causal filtering and are defined bythe difference equations

xL[n - - Mo]

(6)m=0

for — oo < n < —1. The assumption that the switchingoccurs at n = 0 was done purely for convenience. Butclearly these equations hold regardless of when the filtersare switched. However, there is a constraint on how closetogether we can switch our filter. Switching points shouldoccur at least No samples apart. In the next section, weconsider how to determine when to switch a filter and whatcharacteristics the filters should have in order to improvesubjective and objective performance.

3. ADAPTATION CONTROL SYSTEM

The flexibility to switch among a set of different filters ispotentially very powerful. But the key to make this workwell is determining an effective control system for adaptingthe filters relative to the signal. Obviously there are manypossibilities. The one investigated in this work is based onthe observation that filters with good step response charac-teristics tend to perform better in local regions with sharpedges or transitions. Filters with good magnitude responsecharacteristics seem to be better suited for the other re-gions. Thus two sets of filters are used in this system: filterset A, which has good magnitude response characteristics;and filter set B, which has good step response character-istics. Figure 3 shows the spectral magnitude response forthe lowpass filters in sets A and B. Notice that lowpass filterA has good attenuation and transition characteristics whilelowpass filter B does not. Figure 2 shows the step responsefor these filters. Here lowpass filter B has an almost ripplefree characteristic while lowpass filter A does not.

The remaining question is how to switch between thesefilters. Several points within the structure (see Figure 1)can be used to estimate an optimal switch point. For ex-ample, x[n], ro[n], t0[n], or yo[n] represent estimation pointsin the structure that could be used to estimate sharp signaltransition regions. The estimation procedure used here con-sists of computing the first backward difference of the sig-nal taken at one of the estimation points mentioned above.When the magnitude of the first backward difference ex-ceeds a pre-set threshold, we consider this to be a sharptransition and switch to filter set B. When that transitionperiod is over, we switch back to filter set A. It is impor-tant to the performance of the system to switch the filtersprior to and not during the transition. Therefore once atransition point is detected by the first backward differenceestimator, we move back a few samples to the onset of thetransition. Similarly when the estimator falls below the alower threshold, the filter is switched back to filter set A.

In the case where x[n], ro[n], or to[n] are used as es-timation points, side information must be transmitted toindicate the location of the switch point. For natural im-ages, the amount of side information is generally negligible.For example, for the test image Lena, only about 0.0281

110.8

IOj6

•20

40

Q 4D

•W

-100

10 20 30 60•120

03 13 33

U

1

JOJ4

I 0.4

02

10 20 40 SO 6030t (time)

Figure 2: Step response characteristics for the lowpass fil-ters A and B.

03 1 13 2 23 3 33w(ndiin)

Figure 3: Magnitude response characteristics for the low-pass filters A and B.

bits/pixel was consumed by this side information. If, onthe other hand, yo[n] is used as an estimation point, sendingside information can be avoided. More precisely, the estima-tion should be performed on y[n] after it has been coded.This allows the synthesis section to derive the switchingpoint from the exact same signal used by the analysis sec-tion to derive the switching point. This form of the im-plementation is a feedback adaptation approach. Interest-ingly, there is often enough information in the yo[n] signalto make an accurate estimation of major transitions. Evenif the estimation is in error occasionally, the results are notcatastrophic since the system performance is lower boundedby that of conventional subband coding (which is not bad).

4. SIMULATION RESULTS

To evaluate the new analysis/synthesis system, we consid-ered a baseline pyramid two-level system with seven sub-bands and the Enumerative Laplacian Quantization (ELQ)encoder introduced by Darragh [10]. Further, we used themethod of circular convolution to periodically extend ourimage boundaries [6]. The derivation of initial conditionsfor 2-D exact reconstruction with IIR filters is addressedin [6]. The 256 X 256 test image Lena was used in this ex-ample. A section of the original is shown in Figure 4. Figure5 shows Lena coded at 0.4527 bits/pixel with conventionalIIR filter banks. The filters used are recursive (filter setA) with magnitude response and step response plots shownin Figures 3 and 2. The output corresponding to the newsystem is shown in Figure 6. It is coded at the same bit

rate and uses r0[n] as an estimation point. Approximately0.0281 bits/pixel are devoted to the side information asso-ciated with the adaptation. In our feedforward system, weuse an arithmetic coder as a lossless coder for the side infor-mation. The subjective quality is noticably better as shownby examining both coded outputs. The ringing distortion isquite noticeable around the boundary of the hat, as seen in5. On the other hand, the ringing distortion disappeared inthe coded image using the new system, as seen in 6. Littlemagnitude distortion can be seen around the edges sincethe filter set of good step response and bad magnitude re-sponse is allowed to switch back very quickly after only afew samples. This improvement in quality is also reflectedin the PSNR of the coded results.

Several important aspects should be addressed in opti-mizing the performance of the new coder. First, the intelli-gent selection of threshold levels in determining the edges orboundaries within an image is critical. A good edge or leveldetector almost surely guarantees the success of the adapta-tion control system. Secondly, the issue of signal extensionshould be study carefully in light of the new spatially vari-ant system because the derivation of initial conditionslonger depends on only one distinct filter set, but maFinally, a feedback scheme can be considered where side in-formation for edge locations within an image will no longerbe needed apriori for transmission over the channel. Aswith most image coding strategies, much of the improve-ment in quality comes as a result of fine tuning the coder.This has not be done yet at this point but should be donein time for presentation at the conference.

120

Figure 4: Original test image Lena

120

Figure 5: Lena coded at 0.4527 bits/pixel using the conven-tional IIR filter banks with filter A as the basis filters. TheELQ coder is used to code the subband images. (PSNR =28.91 dB)

Figure 6: Lena coded at 0.4527 bits/pixel using the newspatially-varying IIR filter banks. Filters A and B are em-ploying in the switching. The ELQ coder is used to codethe subband images. (PSNR = 29.21 dB)

5. REFERENCES

[1] J. W. Woods and S. D. O'Neil, "Subband coding ofimages," IEEE Trans. Acoutt., Speech, Signal Procen-ing, ASSP-34, pp. 1278-1288, Oct. 1986.

[2] M. J. T. Smith and W. C. Chung, "Adaptive IIRAnalysis-Synthesis Filter Banks," IEEE DSP Work-ihop, September 16-19, 1992, Chicago, Illinois.

[3] M. J. T. Smith and W. C. Chung, "Recursive Time-Varying Filter Banks for Subband Image Coding,"Submitted to IEEE Trans, on Image Processing,

[4] M. J. T. Smith and W. C. Chung, "Spatially AdaptiveIIR Analysis-Synthesis Filter Banks for Image Cod-ing," Proceedings of the Asilomar Conference, Octo-ber, 1992.

[5] Subband Image Coding, (John Woods, Ed.), KluwerAcademic Publishers, Copyright 1991.

[6] M. J. T. Smith and S. L. Eddins, "Analysis/synthesistechniques for subband image coding," IEEE Trans.on ASSP, August 1990.

[7] G. Karlsson and M. Vetterli, "Sub-band coding of finitelength signals," Signal Processing, volume 17, pp. 161-168, June 1989.

[8] J. H. Hnsoy, Subband Coding of Still Imaget and Video,Ph.D. Thesis, University of Trondheim, Norway, 1991.

[9] T. Kronander, Some Aspects of Perception Bated Im-age Coding, Ph.D. Thesis, Linkoping University, Swe-den, 1989.

[10] J. Darragh, Subband and Transform Coding of Images,Ph.D. Thesis, UCLA, 1989.